Well-posedness in Sobolev spaces of the full water wave problem in 3-D

We consider the motion of the interface separating an inviscid, incompressible, irrotational fluid from a region of zero density in three-dimensional space; we assume that the fluid region is below the vacuum, the fluid is under the influence of gravity and the surface tension is zero. Assume that the density of mass of the fluid is one, the gravitational field is (0, 0,−1), the free interface is Σ(t) at time t ≥ 0, and the fluid occupies the region Ω(t). The motion of the fluid is described by vt + v · ∇v = −(0, 0, 1)−∇p on Ω(t), t ≥ 0 (Euler’s equation), (1.1) div v = 0 on Ω(t), t ≥ 0 (incompressible), (1.2) curl v = 0, on Ω(t), t ≥ 0 (irrotational), (1.3) where v = (v1, v2, v3) is the fluid velocity, p is the fluid pressure. Since we neglect the surface tension, the pressure is zero on the interface. So on the interface: p = 0, on Σ(t), (1.4) (1, v) is tangent to the free surface (t, Σ(t)). (1.5) We want to find solutions of system (1.1)-(1.5), taking prescribed initial data, such that for every fixed t ≥ 0, Σ(t) approaches the xy-plane at infinity, and |v(x, y, z; t)| → 0, |vt(x, y, z; t)| → 0, as |(x, y, z)| → ∞. Since the fluid is assumed irrotational, incompressible, we can reduce the study of the entire motion to the motion of the free surface. The above model is a 3-D water wave model. It is generally known that when surface tension is neglected, the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to Taylor instability [8], [22]. In a previous work [24], we studied the 2-dimensional water wave model; we showed that for a 2-D water wave, the sign condition relating to Taylor instability always holds for nonself-intersecting interface, that is, the motion of the interface is not subject to Taylor instability. We showed further that the 2-D full nonlinear water wave problem is uniquely solvable in Sobolev spaces, locally in time, for any initially nonself-intersecting interface. Earlier works on the well-posedness in Sobolev spaces of the 2-D water wave problem include Nalimov [18], Yosihara [25], and Walter Craig [6], where the main results concern the well-posedness of the